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All experimental measurements are inaccurate to some extent. For example,
using a ruler you can only typically measure to an accuracy of a few
millimeters. If you used a micrometer to measure the distance instead, you might
be able to measure the same distance to within a few micrometers.
When writing down a measurement, often scientists will place a +- (number)
after the the measurement to indicate just how uncertain the measurement is. For
example, if you used the above ruler to measure a distance, you might write
0.045 +- 0.001 m, indicating that while you measured a distance of 0.045 meters,
it might be as large as 0.046 or as small as 0.044 meters. For the micrometer,
you will get a much more accurate measurement, and might be able to write the
distance as 0.04522 +- 0.00001 meters, meaning that the distance is between
0.04521 and 0.04523 meters.
One way that scientists use to account for this uncertainty is through the
use of significant figures. Often, they will simply drop the +- and write
0.045 meters instead. It is understood that the last digit has an uncertainty of
at least +- 1. When using this notation, it is very important to keep track of
the number of significant figures in the calculation.
There are several rules which detail how many significant figures a number
has
- All non-zero digits are significant. 3.12 has 3 significant figures,
45.229 has five
- Zeros between non-zero digits are significant. 3.012 has four significant
figures, 45.0009 has six.
- Zeros beyond the decimal point at the end of a number are significant. (Author's
note) 3.340 has four significant figures. Note that there is some
ambiguity in a number such as 500: is there one or three significant figures?
- Zeros preceeding the first non-zero digit are not significant. These are
merely placeholders: 0.0034 has only two significant figures.
- Digits in the exponent in exponential notation are not significant.
1.34*107 has three significant figures (1.34), not 4.
Author's note: The author vastly prefers a simpler form of
rule 3: all trailing zeros are significant. This removes all ambiguity in a
number such as 500: it has three significant figures. If you wanted to write the
number 500 with only one significant figure, use exponential notation:
5*102.
It is important to keep the correct number of significant figures when
adding and
subtracting or multiplying and
dividing numbers.
Example: How many significant figures do the following numbers have:
1.559, 10.4, 0.3400, 0.09970, 1.990*10-5?
Solution:
- 1.559: Simply count the digits. 4 significant figures
- 10.4: The zero is significant: see rule 2. 3 significant figures
- 0.3400: Both trailing zeros are significant: see rule 3. 4 significant
figures
- 0.09970: The leading zeros are not significant, the trailing one is. See
rules 3 and 4. 4 significant figures
- 1.990*10-5: The trailing zero is significant, the -5 in the
exponent is not: see rules 3 and 5. 4 significant figures
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